DRCC-Width-Zeta (DWZ-V2.1): A Width-Controlled Euler–Maclaurin Reconstruction Method for the Riemann Zeta Function in the Critical Strip

Abstract

This work presents a reproducible numerical case study that operationalizes the DRCC/CPR reconstruction-width framework for the computation of the Riemann zeta function \zeta(s) in the critical strip 0<\operatorname{Re}(s)<1. The objective is not to introduce a faster alternative to established methods such as the Riemann–Siegel formula, but to demonstrate how an abstract reconstruction-stability condition can be translated into a concrete, measurable, and testable numerical procedure. The naive truncated Dirichlet sum is shown to diverge in the critical strip as the truncation width increases. A first-order Euler–Maclaurin correction is therefore adopted as the minimal correction within this family that restores convergence in the tested setting. The resulting approximation is called the DRCC-Width-Zeta method, or DWZ. Throughout the manuscript, the numerical width W denotes the truncation depth of the partial sum and is not identified with the structural CPR-Width. An empirically fitted stability estimator is derived for the truncation width required to achieve a prescribed tolerance. On the critical line, the observed zero-position error follows an approximate W^{-3/2} scaling law, consistent with the first omitted Euler–Maclaurin term under the stated local assumptions. The method is checked against mpmath reference values for 100 known nontrivial zeros. Of these, 98 are locally refined from reference starting values, two are recovered through blind scans, and 29 are additionally examined using a hybrid search procedure. The resulting absolute deviations range approximately from 1.5\times10^{-6} to 1.3\times10^{-5}. Comparisons with higher-order Euler–Maclaurin reconstruction and the Riemann–Siegel formula show that the minimal DWZ model is substantially slower and less accurate than established alternatives. This negative performance result is reported explicitly and supports the intended interpretation of DWZ as a methodological operationalization and resource-separation study rather than a competitive zeta algorithm. The work makes no claim concerning a proof of the Riemann Hypothesis and does not identify numerical truncation width with structural CPR-Width.

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